Problem 111 - Calculus Early Transcendentals

Question

Show that the equation \(r=a \cos \theta+b \sin \theta\) where \(a\) and \(b\) are real numbers, describes a circle. Find the center and radius of the circle.

Step-by-Step Solution

Answer

If so, determine the center and radius of the circle. Answer: Yes, the polar equation \(r = a\cos\theta + b\sin\theta\) describes a circle. The center of the circle is \(\left(\frac{ab}{a^2 + b^2}, \frac{b^2 - a^2}{a^2 + b^2}\right)\), and the radius is \(r = \sqrt{h^2 + k^2 + r^2 - h^2 - k^2}\), where \(h = \frac{ab}{a^2 + b^2}\) and \(k = \frac{b^2 - a^2}{a^2 + b^2}\).

Step 1: Simplify the equation using trigonometric identities

To do this, we need to manipulate the equation into the standard form of a circle's equation: \((x-h)^2 + (y-k)^2 = r^2\). Firstly, we square the given equation and use the Pythagorean identity: \(r^2 = a^2\cos^2\theta + b^2\sin^2\theta + 2ab\cos\theta\sin\theta\) since \(\sin^2\theta+\cos^2\theta = 1\).

Step 2: Convert polar form to rectangular form

Keep in mind that \(r^2 = x^2 + y^2\). We use the relationships \(x = r\cos\theta\) and \(y = r\sin\theta\). Substituting these into the equation, we get: \(x^2 + y^2 = a^2(\frac{x^2}{x^2 + y^2}) + b^2(\frac{y^2}{x^2 + y^2}) + 2ab(\frac{x}{x^2 + y^2})\times(\frac{y}{x^2 + y^2})\) Now, we can simplify this equation by multiplying through by \((x^2 + y^2)\): \(x^2(x^2 + y^2) = a^2 x^2 + b^2 y^2 + 2ab xy\)

Step 3: Rearrange the equation to the standard circle equation form

We want to manipulate the equation into the form \((x-h)^2 + (y-k)^2 = r^2\). To do this, complete the square for the x and y terms. The rearranged equation becomes: \((x^2 - 2h x) + (y^2 - 2k y) = r^2 - h^2 - k^2\) where \(h = \frac{ab}{a^2 + b^2}\) and \(k = \frac{b^2 - a^2}{a^2 + b^2}\). Comparing this with the standard circle equation, we find that the center of the circle is \(C(h, k)\) and radius is \(r = \sqrt{h^2 + k^2 + r^2 - h^2 - k^2}\). To conclude, the given equation \(r = a\cos\theta + b\sin\theta\) describes a circle with center \(C(\frac{ab}{a^2 + b^2}, \frac{b^2 - a^2}{a^2 + b^2})\) and radius \(r = \sqrt{h^2 + k^2 + r^2 - h^2 - k^2}\).

Key Concepts

Circle Equation

The circle equation is fundamental in geometry and arises from the more general concept of conic sections. A standard formula for a circle can be expressed as \( (x-h)^2 + (y-k)^2 = r^2 \). In this representation:

• \(x\) and \(y\) denote any arbitrary point on the circle.
• \((h, k)\) is the center of the circle.
• \(r\) is the radius of the circle, representing the constant distance from any point on the circle to its center.

To observe that our equation \( r = a\cos\theta + b\sin\theta \) describes a circle, we eventually aim to convert it into this familiar form. This transformation helps us identify the circle's properties such as the center and radius. Throughout this process, we simplified and rearranged terms to satisfy this standard equation, which offers clarity and insight into the nature of the geometric shape described by the polar equation.

Trigonometric Identities

Trigonometric identities are tools that relate the angles and sides of triangles in a unit circle, often simplifying complex trigonometric expressions. A useful identity for circles is the Pythagorean identity: \( \sin^2\theta + \cos^2\theta = 1 \), which applies to any angle \( \theta \). This identity is among the most fundamental in trigonometry and serves as a cornerstone for deriving various other identities. In this exercise, we used it to transform the polar equation into a squared form, facilitating conversion to rectangular coordinates. This step is crucial for finding the standard circle equation. Additional identities often used include:

• \( \sin(2\theta) = 2\sin\theta \cos\theta \)
• \( \cos(2\theta) = \cos^2\theta - \sin^2\theta \)

These relationships extend our ability to manipulate expressions involving trigonometric functions, enabling a seamless transition between polar and rectangular forms, as illustrated in the problem.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are defined using perpendicular axes traditionally labeled as "x" and "y". In a 2D plane, these coordinates help pinpoint the exact location of a point using a pair \( (x, y) \), where:

• \(x\) represents the horizontal distance from the origin.
• \(y\) represents the vertical distance from the origin.

Transforming polar equations into rectangular coordinates allows us to match the equation with the familiar geometric shapes of Euclidean geometry. The given polar equation \( r = a\cos\theta + b\sin\theta \) provides a path to facilitate its conversion into the Cartesian equation, enabling observation of the shape as a circle.To do this, we use relationships like:

• \( x = r\cos\theta \)
• \( y = r\sin\theta \)

Through substituting these into the original polar equation, we deduced their rectangular analog. This conversion is not only crucial for visualizing geometric objects but also pivotal in recognizing the algebraic reforms of geometric figures.